Finite Element Analysis

Preface......Page 6
Acknowledgement......Page 7
Contents......Page 8
1.2 General Description of the Method\r......Page 14
1.3 A Brief Explanation of FEA for a Stress Analysis Problem\r......Page 15
1.4 Finite Element Method vs Classical Methods\r......Page 17
1.5 FEM vs FDM\r......Page 18
1.7 Need for Studying FEM\r......Page 19
References\r......Page 20
2.2 Stresses in a Typical Element \r......Page 22
2.3 Equations of Equilibrium\r......Page 25
2.5 Strain Displacement Equations\r......Page 26
2.6 Linear Constitutive Law......Page 27
Questions\r......Page 32
3.2 Matrix Displacement Equations\r......Page 34
3.3 Solution of Matrix Displacement Equations\r......Page 41
3.4 Techniques of Saving Computer Memory Requirements\r......Page 43
Questions\r......Page 45
4.2 Element Shapes\r......Page 46
4.3 Nodes\r......Page 51
4.4 Nodal Unknowns\r......Page 52
4.5 Coordinate Systems\r......Page 53
Questions\r......Page 66
5.1 Introduction\r......Page 68
5.2 Polynomial Shape Functions\r......Page 69
5.3 Convergence Requirements of Shape Functions\r......Page 71
5.4 Derivation of Shape Functions Using Polynomials\r......Page 74
5.5 Finding Shape Functions Using Lagrange Polynomials\r......Page 95
5.6 Shape Functions for Serendipity Family Elements\r......Page 102
5.7 Hermite Polynomials as Shape Functions \r......Page 108
5.8 Construction of Shape Functions by Degrading Technique\r......Page 111
Questions\r......Page 115
6.2 Strain-Displacement Matrix for Bar Elements\r......Page 117
6.3 Strain Displacement Matrix for CST Element\r......Page 118
6.4 Strain Displacement Matrix for  a Beam Element\r......Page 120
Questions\r......Page 121
7.2 Element Stiffness Matrix for CST Element by Direct Approach\r......Page 123
7.3 Nodal Loads by Direct Approach\r......Page 127
Questions\r......Page 130
8.2 Galerkin\'s Method\r......Page 131
8.3 Galerkin\'s Method Applied to Elasticity Problems\r......Page 132
Questions\r......Page 140
9.2 General Variational Method in Elasticity Problems\r......Page 141
9.3 Potential Energy in Elastic Bodies\r......Page 147
9.4 Principle of Minimum Potential Energy\r......Page 149
9.5 The Rayleigh-Ritz Method\r......Page 153
9.6 Variational Formulation in Finite Element Analysis\r......Page 163
Questions\r......Page 166
10.2 Nodes at Discontinuities\r......Page 167
10.3 Refining Mesh\r......Page 169
10.5 Finite Representation of Infinite Bodies \r......Page 170
10.6 Element Aspect Ratio\r......Page 171
10.8 Numbering System to Reduce Band Width\r......Page 172
Questions\r......Page 173
11.2 Tension Bars/Columns\r......Page 174
11.3 Two Dimensional Trusses (Plane Trusses)\r......Page 193
11.4 Three Dimensional Trusses (Space Trusses)\r......Page 210
Questions\r......Page 214
12.2 General Procedure when CST Elements are Used\r......Page 217
12.3 Use of Higher Order Elements\r......Page 229
Questions\r......Page 230
13.1 Introduction\r......Page 232
13.2 Coordinate Transformation\r......Page 234
13.3 Basic Theorems of Isoparametric Concept\r......Page 235
13.4 Uniqueness of Mapping\r......Page 236
13.5 Isoparametric, Superparametric and Subparametric Elements\r......Page 237
13.6 Assembling Stiffness Matrix\r......Page 238
13.7 Numerical Integration\r......Page 243
13.8 Numerical Examples\r......Page 245
Questions\r......Page 253
References\r......Page 254
14.2 Beam Analysis Using Two Noded Elements\r......Page 255
14.3 Analysis of Rigid Plane Frames Using 2 Noded Beam Elements\r......Page 272
14.4 A Three Dimensional Rigid Frame Element\r......Page 279
14.5 Timoshenko Beam Element\r......Page 282
Questions\r......Page 291
References\r......Page 292
15.1 Introduction\r......Page 293
15.2 Basic Relations in Thin Plate Theory\r......Page 294
15.3 Displacement Models for Plate Analysis\r......Page 295
15.4 Rectangular Plate Element with 12 Degrees of Freedom\r......Page 297
15.5 Rectangular Plate Element with 16 Degrees of Freedom\r......Page 302
15.6 Mindlin\'s Plate Element\r......Page 305
References\r......Page 312
16.2 Forces on Shell Element\r......Page 314
16.3 Finite Elements of Shell Analysis\r......Page 315
16.4 Finite Element Formulation Using Four Noded Degenerated Quadrilateral Shell Element\r......Page 320
References\r......Page 330
17.2 Non-Linear Problems\r......Page 331
17.3 Analysis of Material Non-Linear Problems\r......Page 333
17.4 Analysis of Geometric Non-Linear Problems \r......Page 338
References\r......Page 341
18.2 Commercially Available Standard Packages\r......Page 342
18.3 Structure of a Finite Element Analysis Program\r......Page 343
18.4 Pre and Post Processors\r......Page 344
Questions\r......Page 346